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Chapter 4 · Another Peek Beyond the Point

Dividing when the dividend has a decimal

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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

  • Divide a decimal by 10, 100 or 1000 and say where the point ends up
  • Explain the point-shifting rule from what division does to each place value, rather than asserting it
  • Work a division-by-powers-of-ten table backwards: given a result, recover the starting decimal
  • Divide a decimal dividend by a whole number using long division, placing the point correctly
  • Handle a dividend smaller than 1, where the quotient opens with zeros
  • Say what a zero in the Tenths place of a quotient is recording
  • Predict how a quotient changes when the dividend is divided by ten but the divisor is left alone

Where it usually goes wrong

  • "When the dividend has a point, put the point in the answer above it." That recipe happens to work for a whole-number divisor and gives no reason why. Ask instead which step regrouped Ones into Tenths — the answer is the same and it survives into the next topic, where the divisor has a point and the recipe breaks.
  • "Dividing by 10 moves the digits, and dividing by 0.1 also moves the digits, so the direction does not matter." The chapter's rule is explicitly leftward for division by 10, 100 and 1000. Which way it goes is the whole content of the rule, and Q4 on Part II p.87 asks the student to fill in a divisor that produces a larger answer.
  • "A zero in the quotient means nothing happened, so it can be left out." 0.06 divided by 5 has a zero in the Tenths place recording that no Tenths could be handed out. Delete it and the answer is ten times too big.
  • "You need a new method once the dividend has a point." The subheading announcing a decimal dividend is followed by two examples worked in exactly the layout of the previous four pages. The novelty is in the dividend, not in the method.
  • "3.9 divided by 100 has two decimal places, because 100 has two zeros." It has three. Counting zeros tells you how far the point moves, not how many places the answer ends up with — the dividend's own places are still there underneath.
  • "Dividing always makes a number smaller, so 0.06 divided by 5 must be nearly nothing." It is smaller here, because 5 is bigger than 1. The general claim is taken apart in Dividing when the divisor has a decimal, and it should not be reinforced here.

Questions to check understanding

  • Divide a given decimal by 10, 100 and 1000, and complete a table of such results
  • Recover a starting decimal from one of its divided results
  • Divide a decimal by a whole number by long division, to a stated number of places
  • Fill a blank divisor so that a stated quotient comes out (the shape of Q4 on Part II p.87)
  • Word problems: a weight split into equal packets, a length cut into equal pieces, a quantity shared among a stated number of people
  • Explain what a particular zero in a quotient is recording
  • Convert between units by dividing by a power of ten — the shape of Q5 on Part II p.94

Examples worth working on the board

  • Example 6, Anuja's ribbon (Part II, §4.3, p.74). Checked against the printed page. A full-colour picture fills the right of the page: a girl at a brown table with scissors and a red ribbon, a double-headed arrow beneath the ribbon labelled 3.9 m, and a thought bubble above her holding a row of ten small red rectangles and the question of how long each piece is. Inputs: a 3.9 m ribbon and 10 equal pieces. The book then repeats the question with 100 pieces.
  • The reciprocal step (Part II, §4.3, p.74). The book turns 3.9 into 39 over 10, then divides by 10 by multiplying by one-tenth.
  • The division table (Part II, §4.3, p.75). Checked against the printed page. Five rows and five columns, headed Decimal, ÷ 10, ÷ 100, ÷ 1000 and ÷ 10000. The first row is fully worked from 18.7. The next two rows give only a starting decimal — 21.1 and 0.13 — and leave the rest blank. The last two rows give no starting decimal: one supplies only its ÷ 100 entry, 2.146, and the other only its ÷ 10000 entry, 0.0058. Those two rows have to be run backwards, and that is what makes the table worth showing.
  • The whole-number rule (Part II, §4.1, p.68). Checked against the printed page. Three numbered steps for 123 ÷ 10 — write the dividend with a point at its right-hand end, count the zeros in the divisor, step the point left by that many places, padding with zeros in front if needed — with a small blue arrow curling back under the answer to show the direction of travel. Four worked instances follow on p.69 in a two-by-two box: 24 ÷ 100, 678 ÷ 1000, 12 ÷ 1000 and 12345 ÷ 1000, each with its own curling arrow and its new zeros set in red.
  • Example 10, the sugar (Part II, §4.3, p.82). Checked against the printed page. 9.5 kg packed equally into 4 bags, worked as a single long division whose column labels read O, Tenths, Hundredths, Thousandths — note that there is no Tens column at all, because the dividend has none. Inputs: 9.5 and 4.
  • Example 11, the small dividend (Part II, §4.3, p.83). Checked against the printed page. 0.06 divided by 5, broken out in words as zero Ones, zero Tenths and six Hundredths before the division starts, and worked in a layout whose quotient begins with a zero, a point, and then another zero. Inputs: 0.06 and 5.
  • "Figure it Out" Q3 (Part II, §4.3, p.83). The divisor is 4 throughout, and the dividend runs 132, then 13.2, then 1.32, then 0.132. Four quotients, each a tenth of the one above it.
  • "Figure it Out" Q4 (Part II, §4.3, p.83). The same idea with divisor 8 and five dividends: 126, 12.6, 1.26, 0.126 and 0.0126. These quotients run much longer than Q3's — up to six decimal places against three — which is worth noticing out loud.
  • The reminder cartoon (Part II, §4.3, p.83). Checked against the printed page. A girl in a blue jacket in the right margin with a green speech bubble restating that the quotient takes its point when Ones become Tenths. The same reminder is printed on p.81 with a different cartoon, and again in body type on pp.81, 82 and 83 — the chapter keeps saying it.

Figures to have open

  • A ribbon of stated length with adjustable cut marks, so ten pieces and a hundred pieces can both be shown. Standard schematic.
  • A place-value strip showing a decimal's digits with a movable point, for the shift rule. This is the topic's core image and must be reusable in sections 4, 6 and 11.
  • A five-by-five grid for the division table, fillable in any order.
  • The Indian long-division layout with vertical colour-coded place labels above the quotient — matching the book's layout, redrawn.
  • 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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