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Chapter 4 · Another Peek Beyond the Point

Dividing when the divisor has a decimal

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11 min.

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What the video covers, what to say before it, where a class usually goes wrong, and what to set afterwards. An account is free, and it opens every chapter of every book.

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What to assume they know

What they should be able to do

  • Convert a division with a decimal divisor into one with a whole-number divisor
  • Say which power of ten to scale by, from the number of places in the divisor
  • Justify the scaling by equivalence of fractions rather than by a memorised shift
  • Compute a quotient with a decimal divisor and interpret it in context, such as an average speed
  • State when a quotient exceeds its dividend, and when it falls below
  • Identify the divisor for which quotient and dividend are equal, and explain why the chapter's opening sentence does not cover it
  • Derive a family of quotients from a single known whole-number division
  • Recognise two differently written divisions as the same division

Where it usually goes wrong

  • "Move the point in the divisor and leave the dividend alone." This is the commonest wrong version of the rule and it changes the answer by a factor of ten. The chapter's justification blocks it: what makes the move legal is that the fraction is unchanged, and a fraction is only unchanged when both parts are multiplied.
  • "Dividing always makes a number smaller." 128 divided by four-tenths is 320. The direction is decided by whether the divisor sits above or below 1, exactly as it was for multiplication in Why multiplying by a decimal below 1 shrinks a number.
  • "The quotient of two counting numbers is always smaller than the dividend." The chapter opens the Part II p.86 subheading with this and it is not quite true — dividing by 1 returns the dividend unchanged. Say so; it is a one-line fix and it is the same boundary case that the multiplication table on Part II p.72 also leaves out.
  • "Scaling by 10 always works, so use 10 every time." The power of ten needed is fixed by the divisor's decimal places: 1.3 needs ten, 0.13 needs a hundred. Example 13 is printed as a pair precisely so this can be seen.
  • "24.6 divided by 1.5 and 2.46 divided by 0.15 are different problems because the numbers look different." They are the same division written twice, and Q5 on Part II p.87 asks about exactly this pair.
  • "Dividing by a decimal is a special rule to be memorised alongside the others." It is a way of getting back to the only division rule there is. The chapter's own sentence after Example 13 says the whole point is to reach a counting-number divisor and then carry on as before.

Questions to check understanding

  • Compute a quotient whose divisor carries one, two or three decimal places
  • Given one whole-number division, write down a family of rescaled quotients (the shape of Q8 on Part II p.94 and Q3 on Part II p.87)
  • Complete a two-way grid of quotients from a few given cells (Q9 on Part II p.94)
  • Fill a blank so that a stated quotient results, including blanks where a division has to be rewritten as a multiplication (Q4 on Part II p.87)
  • Average speed, distance per litre, and cost per unit — all three appear as word problems in this chapter
  • Given a regular polygon's perimeter and how many sides it has, find one side
  • Decide whether a quotient will be larger or smaller than the dividend before computing it

Examples worth working on the board

  • Example 12, the scooter (Part II, §4.3, p.84). Ravi rides from Pune to Matheran, 126 km, in 2.5 hours, and his average speed is wanted. The book rewrites 2.5 as 25 over 10, inverts it, and reaches a division of 1260 by 25. It then states the quotient without showing the long division, which leaves the explanation a step to fill in.
  • Example 13, the paired divisions (Part II, §4.3, p.84). 4.68 divided by 1.3, then 4.68 divided by 0.13. The same dividend, the same digits in the divisor, one extra decimal place. The book carries both as far as a whole-number divisor of 13 and stops.
  • The generalised statement (Part II, §4.3, p.84). The chapter sets 4.68 over 0.13 beside the same fraction with both parts multiplied by 100, ending at 468 over 13. Three expressions on one line. This is the figure to redraw.
  • The 128 pair (Part II, §4.3, p.86). Checked against the printed page. 128 divided by 4 gives 32; 128 divided by 0.4 gives 320. The book prints both, side by side in the running text, and then asks whether a decimal divisor always pushes the quotient above the dividend, with the marginal Math Talk flag beside the question.
  • The table the reader is asked to make (Part II, §4.3, p.86). Checked against the printed page. The chapter asks for a table of the relationship between dividend, divisor and quotient, on the model of the multiplication table on p.72 — and it prints no such table. Nothing between the request on p.86 and the SUMMARY on p.95 supplies one; checked on the printed page. The explanation is therefore building it, and must say so.
  • "Figure it Out" Q5 (Part II, §4.3, p.87). 2.46 divided by 1.5, by 0.15 and by 0.015, followed by a question asking whether 24.6 divided by 1.5 gives the same quotient as 2.46 divided by 0.15.
  • "Figure it Out" Q3 (Part II, §4.3, p.87). Given 156 × 12 = 1872, four results are asked for, two multiplications and two divisions — including one that divides by a decimal. It is the same derive-from-one-known-fact move used for multiplication in Multiply as whole numbers, then count the decimal digits.
  • "Figure it Out" Q4 (Part II, §4.3, p.87). Six blanks, the last three of which ask the student to rewrite a division of 25 as a multiplication. Three items, three divisors — 10, then 0.10, then 0.01 — and they do not all pull the same way: one of them turns into multiplying by something below 1 while the other two turn into multiplying by 10 and by 100. Name all three; dropping the divisor of 10 loses the only case where the multiplier comes out smaller than 1. This is where the inverse relationship is meant to click.
  • "Figure it Out" Q8 and Q9 (Part II, §4.4, p.94). Checked against the printed page. Q8 gives 756 ÷ 36 = 21 and asks for six rescaled quotients. Q9 is a five-by-five grid whose column heads are 1517, 151.7, 15.17, 1.517 and 15170 and whose row heads are 37, 3.7, 0.37, 0.037 and 370, with a small arrow diagram in the corner cell defining every entry as the column value divided by the row value; three cells are pre-filled — 41, 4.1 and 4100 — and the rest are blank. This grid is the best single exercise in the chapter for this topic.

Figures to have open

  • The Indian long-division layout with a decimal divisor in place, so the blocking step can be shown and then removed. Redraw it.
  • A three-term equality line — original fraction, scaled fraction, whole-number divisor — with the scaling factor shown on both parts. Load-bearing; this is the argument.
  • A number line long enough to carry 32 and 320 against a dividend of 128.
  • A branching diagram for divisor above 1, equal to 1, and between 0 and 1.
  • A five-by-five grid with the row and column heads of Q9, fillable cell by cell.
  • No photograph and no data set from the textbook is required for this topic.

Where this sits in the book

The book

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