Divisions that never end — teaching notes

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What to assume they know

Long division continued past the ones place — long division as regrouping, and the remainder at each step
Dividing when the dividend has a decimal — the quotient's point, and quotients that open with zeros
Knowing that a remainder is always smaller than the divisor
Breaking a number down to primes, and why the result is unique — prime factorisation, needed for the closing pattern
Conjecture and generalisation: what mathematicians mean by those words — what a conjecture is, and that some are still open

What they should be able to do

Carry out a long division far enough to see a remainder come round again
Explain, without appeal to a calculator, why some divisions cannot terminate
Give an upper bound on how long a repeating block can be, from the divisor alone
Trace the chain of remainders for one divided by seven and relate it to the quotient's digits
Describe what happens when 142857 is multiplied by the numbers up to seven
State the 1927 conjecture the chapter mentions and say why it is still not a theorem
Say which denominators do give a terminating decimal, and connect that to the factors of ten

Where it usually goes wrong

"It never ends because the calculator ran out of room." The figure is irrelevant. The book's argument is about the leftover: at every step it is one, and one behaves the same way every time it appears.
"It never ends, so nobody knows what the digits are." The digits are completely determined and the pattern is short. Not ending and not being known are different things.
"You could get lucky and have it stop later on." No — once a leftover recurs, the working from that point is an exact copy of the working from the first occurrence, so it is committed for ever. This is the step the chapter observes and does not prove, and it is what section 8 supplies.
"Every division that does not stop repeats after one step, like ten over three." Ten over three repeats with a block of one digit; one over seven needs six; one over seventeen needs many more. The length varies, and it is bounded by the divisor, not fixed by it.
"142857 is magic." It is the repeating block of one over seven, and the rotations are the same six digits started at a different point of the cycle. Everything striking about it comes from the cycle the explanation has just drawn.
"A conjecture is a fact that has not been written down yet." The 1927 conjecture on Part II p.86 has resisted a century of work. Use it as the book does — as evidence that mathematics has open frontiers reachable from Class 7 arithmetic.
"Decimals that stop and decimals that repeat are two unrelated kinds." The panel on Part II p.87 is the bridge: it is the divisor's factors that decide, and ten is built from two and five.

Questions to check understanding

Divide two given whole numbers far enough to identify the repeating block
State the greatest possible length of a repeating block for a stated divisor
Say whether a given unit fraction gives a decimal that stops, and justify it from the denominator's factors
Express fractions such as two-fifths, thirteen-quarters and five-eighths in decimal form — the shape of Q1 on Part II p.86
Explain in words why a stated division cannot terminate
Multiply a cyclic number by small whole numbers and describe what happens
Distinguish a conjecture from a proved result

Examples worth working on the board

Ten divided by three (Part II, §4.3, pp.84–85). Checked against the printed page. The book sets it out as four numbered steps in the left column — regroup one Ten into ten Ones, then one One into ten Tenths, then one Tenth into ten Hundredths, then one Hundredth into ten Thousandths and one Thousandth into ten TenThousandths — with the long division standing in the right margin, its quotient trailing off in dots and the pairs 10 over 9 stacking down the page. Inputs: 10 and 3.
Two more to try (Part II, §4.3, p.85). Ten divided by nine, and a hundred divided by eleven. Both are set as questions and neither is worked. They are useful because their repeating blocks are of different lengths, one digit and two.
One divided by seven (Part II, §4.3, p.85). Checked against the printed page. The long division runs the full height of the right margin, twenty-odd lines deep, and two of the leftovers in it are ringed in red — the first 10 and the final 1 — to mark where the working comes back to where it began. Inputs: 1 and 7. Do not pre-supply the six digits.
The chain diagram (Part II, §4.3, p.85). Checked against the printed page. Three short rows of numbers joined by arrows, laid out boustrophedon so the chain snakes: the first row runs left to right and its arrows point right, the second runs right to left, and a third row starts a third pass and breaks off in dots. The numbers in it are the successive leftovers, and the same six values come round twice before the diagram gives up. Redraw it as a ring rather than as rows — the ring is what the layout is trying to say.
142857 (Part II, §4.3, pp.85–86). The reader is asked to multiply it by each of 1, 2, 3, 4, 5 and 6, and then separately by 7.
One divided by seventeen (Part II, §4.3, p.86). Set as a Try This: find it in decimal and use the repeating block. The block is much longer than seven's, which is the point.
The terminating panel (Part II, §4.3, p.87). Checked against the printed page. A boxed two-column panel at the foot of the page. The left column runs one over two, one over two-times-two, one over two-times-two-times-two, and one more, with the decimals 0.5, 0.25, 0.125 and 0.0625 given, and a fifth line whose value is left as a question mark. The right column does the same with fives: 0.2, 0.04, 0.008, 0.0016, and a fifth left open. The denominators are deliberately printed as products rather than as 4, 8, 16, so the factors are visible. A Math Talk question underneath asks what the pattern is and why two and five should be related this way.

Figures to have open

A long-division layout that can be grown line by line with the remainder highlighted at each stage. Redraw the book's Indian layout.
A ring of six positions carrying the leftovers of one divided by seven, with a travelling marker. This is the topic's central image and the book's own row layout is only a flattened version of it.
A second concentric ring of quotient digits, so the two cycles can be shown turning together.
A stacked comparison of 142857 and its multiples, aligned so the rotation is visible.
A two-column panel of unit fractions with denominators written as products, for section 11. Redraw; do not lift the printed panel.
No photograph and no data set from the textbook is required for this topic.

Where this sits in the book

NCERT Ganita Prakash, Class 7, Part II, printed Chapter 4 "Another Peek Beyond the Point", §4.3 "Decimal Division", the subheading "Does This Ever End?", pp.84–85 — the four-step working of ten divided by three, the two further divisions set as questions, one divided by seven, and the chain diagram
Same part, same chapter, §4.3, the subheading "A Magic Number: 142857", pp.85–86 — the multiplications, the Try This on one divided by seventeen, and the 1927 conjecture
Same part, same chapter, §4.3, p.87 — the boxed panel of powers of two and powers of five, and the question printed under it
Same part, same chapter, §4.3, "Figure it Out", p.86 — question 1
Same part, same chapter, SUMMARY, p.95, bullet 4
Backward pointers: Long division continued past the ones place, Breaking a number down to primes, and why the result is unique and Conjecture and generalisation: what mathematicians mean by those words