4. Another Peek Beyond the Point

7 topics1 h

Chapter 4 of NCERT Mathematics for Class 7: Another Peek Beyond the Point. Three sections — Multiplying decimals, Dividing decimals and Checking the answer before you trust it. Seven videos, 1 h in all.

Worked answers to the 35 exercise questions in this chapter →

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Part 2

Multiplying decimals

11 minMultiply as whole numbers, then count the decimal digitsMultiply the digits, then count the decimal places and add them. Everybody is taught the instruction; almost nobody is told why it is allowed.11 min · 12 partsहिंदी10 minWhy multiplying by a decimal below 1 shrinks a number“Multiplying makes things bigger” is the first rule anybody learns about multiplication, and it is false.10 min · 12 partsहिंदी

Part 3

Dividing decimals

11 minLong division continued past the ones placeWhere does the point go in a division answer? Almost everybody is taught a rule, and almost everybody forgets it under pressure.11 min · 12 partsहिंदी10 minDividing when the dividend has a decimalEverybody can divide by ten. Almost nobody can say why moving the point one place left is the right thing to do.10 min · 12 partsहिंदी11 minDividing when the divisor has a decimalA divisor with a point in it is the one case long division cannot even begin. Not stall partway — begin.11 min · 12 partsहिंदी10 minDivisions that never endA division that never finishes is not a failure of patience. You can settle it long before you get bored.10 min · 12 partsहिंदी

Part 4

Checking the answer before you trust it

10 minEstimating first, so a wrong answer is visibleOne circuit of the Sun takes 365.2422 days and a calendar counts 365. That leftover fifth of a day is why February has a spare one.10 min · 12 partsहिंदी

What you will be able to do

  • Convert a decimal into a whole number over a power of ten, and back
  • State, before multiplying, whether a product will exceed both factors, fall below both, or land between them
  • Turn a fraction into a decimal by finding an equivalent fraction whose denominator is a power of ten, and say when that is possible
  • Divide a decimal by 10, 100 or 1000 and say where the point ends up
  • Convert a division with a decimal divisor into one with a whole-number divisor
  • Carry out a long division far enough to see a remainder come round again
  • Explain why a small decimal cannot be dismissed until you know what it is multiplied by

What this chapter assumes you already know

  • Multiplying two whole numbers of three or four digits by hand
  • Knowing that 10 × 10 = 100 and 10 × 100 = 1000, so a product of powers of ten is again 1 followed by zeros
  • Comparing two decimals and saying which is larger
  • Knowing that 0.75 and three-quarters are the same quantity, so that "a part of" is a reading available for a decimal too
  • Reading a table with a situation column and an outcome column
  • Whole-number long division with a remainder, including the regrouping that turns a leftover Hundred into ten Tens
  • Finding an equivalent fraction by multiplying numerator and denominator by the same number
  • Knowing which small numbers divide 10, 100 and 1000

It builds directly on Extending Indian place value to the right of the point, A fraction of a fraction, and why the numerators and denominators multiply, Why multiplying can make a number smaller, Reciprocals, and Brahmagupta's rule for dividing fractions, Why dividing can make a number bigger and Breaking a number down to primes, and why the result is unique.

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.

  • "Line the decimal points up, the way you do when adding."
  • "The product has as many decimal places as the longer factor."
  • "Adding a trailing zero changes the number."
  • "The rule is arbitrary, so it has to be memorised."
  • "596 × 248 has six digits, so 5.96 × 24.8 must too."
  • "You must convert to fractions every time, or it is not proper working."
  • "Multiplication always makes a number bigger."
  • "If any factor is less than 1, the product is less than 1."